Quantum Unipotent Subgroup and dual canonical basis

TL;DR

This paper explores the quantization of cluster algebra structures on unipotent subgroup coordinate rings, relating quantum cluster algebras to dual canonical bases and extending known results to general Kac-Moody algebras.

Contribution

It introduces a framework for quantizing cluster algebra structures and proposes a conjecture linking quantum cluster algebras to dual canonical bases, generalizing previous ADE cases.

Findings

Quantum analogue $O_q[N(w)]$ contains a basis from ${B}^{up}$

The basis includes quantum flag minors and has a factorization property

Generalizes Caldero's results to arbitrary symmetrizable Kac-Moody Lie algebras

Abstract

Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical basis $S^{*}$. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras to the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$, which contains quantum flag minors and satisfies a factorization property with respect to the `$q$-center' of $O_{q}[N(w)]$. This generalizes Caldero's results from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.